They blame ill-health - including cancer, skin defects and fertility problems - on their involvement in British nuclear tests in Australia, on Christmas Island and in the Pacific Ocean between 1952 and 1958.

The MoD acknowledges a "debt of gratitude" but denies negligence.

More than 1,000 veterans want compensation and have been fighting for permission to launch damages claims for more than two years.

In 2009, 10 "lead" claimants won the first round of the battle when a High Court judge said claims could go ahead. But the MoD appealed, and last year the Court of Appeal ruled against veterans - blocking nine of the 10 "lead" claims after appeal judges said they were "statute-barred" because they had been made too late.

Appeal judges also declined to exercise their powers of discretion because they said veterans lacked evidence about the causes of illnesses.

Lawyers representing veterans will try to persuade a panel of seven Supreme Court justices to overrule the Court of Appeal and allow all 10 "lead" claims to go ahead. If they succeed, damages claims would probably be launched in the High Court.

The Supreme Court hearing is due to last three days but the panel - made up of Lord Phillips, Lord Walker, Lady Hale, Lord Brown, Lord Mance, Lord Kerr and Lord Wilson - is likely to reserve judgment to a later date.

Veterans were given permission to appeal to the Supreme Court following a hearing in July. After that hearing, Lord Phillips, president of the Supreme Court, warned ex-servicemen: "The court would not wish to raise false optimism in what are obviously very difficult cases."